Properties

Label 4-585e2-1.1-c1e2-0-30
Degree $4$
Conductor $342225$
Sign $1$
Analytic cond. $21.8205$
Root an. cond. $2.16130$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3·3-s + 2·4-s + 5-s + 3·6-s + 7-s + 5·8-s + 6·9-s + 10-s − 2·11-s + 6·12-s + 13-s + 14-s + 3·15-s + 5·16-s − 8·17-s + 6·18-s + 2·20-s + 3·21-s − 2·22-s + 3·23-s + 15·24-s + 26-s + 9·27-s + 2·28-s + 29-s + 3·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s + 4-s + 0.447·5-s + 1.22·6-s + 0.377·7-s + 1.76·8-s + 2·9-s + 0.316·10-s − 0.603·11-s + 1.73·12-s + 0.277·13-s + 0.267·14-s + 0.774·15-s + 5/4·16-s − 1.94·17-s + 1.41·18-s + 0.447·20-s + 0.654·21-s − 0.426·22-s + 0.625·23-s + 3.06·24-s + 0.196·26-s + 1.73·27-s + 0.377·28-s + 0.185·29-s + 0.547·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(342225\)    =    \(3^{4} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(21.8205\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{585} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 342225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.274377097\)
\(L(\frac12)\) \(\approx\) \(7.274377097\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2$ \( 1 - p T + p T^{2} \)
5$C_2$ \( 1 - T + T^{2} \)
13$C_2$ \( 1 - T + T^{2} \)
good2$C_2^2$ \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 13 T + 122 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2^2$ \( 1 - T - 66 T^{2} - p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 11 T + 38 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.79821951436310820075465847427, −10.66533307081293798832379055621, −9.905156020513125021216327826993, −9.762232686301128468934122640461, −8.872052845965269038322760928171, −8.865223793065681454817506951698, −8.267573270072988521579491972928, −7.71580590381207353225188753825, −7.29581762595020488631745951599, −7.20963405022649932066197688874, −6.26834258761437376344374484753, −6.18656204873664778335274499982, −5.14691771028516477616526773907, −4.57754782812292633278420179170, −4.48588216180960715300826204209, −3.68305172604854356812886061121, −3.01130158522436607329228570294, −2.60085713749009244616324541669, −1.75268678312710092264909982369, −1.72194128531540891089349239436, 1.72194128531540891089349239436, 1.75268678312710092264909982369, 2.60085713749009244616324541669, 3.01130158522436607329228570294, 3.68305172604854356812886061121, 4.48588216180960715300826204209, 4.57754782812292633278420179170, 5.14691771028516477616526773907, 6.18656204873664778335274499982, 6.26834258761437376344374484753, 7.20963405022649932066197688874, 7.29581762595020488631745951599, 7.71580590381207353225188753825, 8.267573270072988521579491972928, 8.865223793065681454817506951698, 8.872052845965269038322760928171, 9.762232686301128468934122640461, 9.905156020513125021216327826993, 10.66533307081293798832379055621, 10.79821951436310820075465847427

Graph of the $Z$-function along the critical line